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Question: The density of copper is \(8.95 g cm^-3\). It has a face-centered cubic structure. What is the radiu...

The density of copper is 8.95gcm−38.95 g cm^-3. It has a face-centered cubic structure. What is the radius of the copper atom? (Atomic mass of Cu = 63.5gmol−163.5 g mol^{-1}, NA=6.022X1023mol−1N_A = 6.022 X 10^{23} mol^{-1})

Explanation

Solution

As we know that the face centred cubic unit cell the atoms are arranged at each corner of the unit along with one atom at the centre of each square. The atoms at the corner are shared by 8 such units and the atom at the face is shared by two units. Total number of atoms in one cell is 4 atoms.

Complete step by step answer:
Let’s start with discussing the face centred cubic structure for better understanding of the question. In a face centred cubic unit cell the atoms are arranged at each corner of the unit along with one atom at the centre of each square. The atoms at the corner are shared by 8 such units and the atom at the face is shared by two units. Hence, the total number of atoms in one face centred cubic unit cell is 88+62=1+3=4\dfrac{8}{8} + \dfrac{6}{2} = 1 + 3 = 4 atoms (8 atoms at corner + 6 atoms at faces). Also the volume of the unit cell is given using 88r38\sqrt {8{r^3}} .
Now coming back to question, we are given the atomic mass of copper, from this we can find out the mass of 1 atom.
Mass of 1 Cu atom = Molecular Mass of CuAvogadro’s Number\dfrac{{{\text{Molecular Mass of Cu}}}}{{{\text{Avogadro's Number}}}}= 63.5 g mol−16.022×1023 mol−1=1.055×10−22g\dfrac{{63.5{\text{ g mo}}{{\text{l}}^{ - 1}}}}{{6.022 \times {{10}^{23}}{\text{ mo}}{{\text{l}}^{ - 1}}}} = 1.055 \times {10^{ - 22}}g
Similarly, Mass of one face centred cubic structure of Cu =4×1.055×10−23g4 \times 1.055 \times {10^{ - 23}}g
Now the volume of one cubic unit cell will be given by
Mass of CuDensity=4×1.055×10−228.95=4.71×10−23cm3\dfrac{{{\text{Mass of Cu}}}}{{{\text{Density}}}} = \dfrac{{4 \times 1.055 \times {{10}^{ - 22}}}}{{8.95}} = 4.71 \times {10^{ - 23}}{\text{c}}{{\text{m}}^3}
We know the formula for Volume, which is V = 88r38\sqrt {8{r^3}}
4.71×10−234.71 \times {10^{ - 23}} = 88r38\sqrt {8{r^3}}
r3r^3 = 4.71×10−2388\dfrac{{4.71 \times {{10}^{ - 23}}}}{{8\sqrt 8 }}= 2.07×10−24cm2.07 \times {10^{ - 24}}cm
Therefore, the radius of Copper atom is 1.28×10−24cm1.28 \times {10^{ - 24}}cm

Note:
We also know that the face centred cubic unit cell is the most packed cubic unit cell. Other than this we are having a Body centred and simple Cubic unit cell. The least packed configuration with most void spaces is Simple cubic unit cell. It contains only 1 atom whereas in a Body centred cubic unit cell, it contains 2 atoms.